1st 2 sections of theory OK

BSE-PES.tex

@@ -36,7 +36,6 @@
 \newcommand{\QP}{\textsc{quantum package}}
 \newcommand{\T}[1]{#1^{\intercal}}
 
-
 % coordinates
 \newcommand{\br}[1]{\mathbf{r}_{#1}}
 \newcommand{\dbr}[1]{d\br{#1}}
@@ -51,6 +50,11 @@
 \newcommand{\Ha}{\text{H}}
 \newcommand{\co}{\text{x}}
 
+% 
+\newcommand{\Norb}{N}
+\newcommand{\Nocc}{O}
+\newcommand{\Nvir}{V}
+
 % operators
 \newcommand{\hH}{\Hat{H}}
 
@@ -271,7 +275,7 @@ where $\W{}$ is the screened Coulomb operator, and hence the BSE reduces to
 	\XiBSE{}(3,5,4,6) = \delta(3,4) \delta(5,6) \vc{}(3,6) - \delta(3,6) \delta(4,5) \W{}(3,4),
 \end{equation}
 where, as commonly done, we have neglected the term $\delta \W{}/\delta \G{}$ in the functional derivative of the self-energy. \cite{Hanke_1980,Strinati_1984,Strinati_1982}
-Finally, the static approximation is enforced, \ie, $\W{}(1,2) = \W{}(\{\br{1}, \sigma_1, t_1\},\{\br{2}, \sigma_2, t_2\}) \delta(t_1-t_2)$ which corresponds to restricting $\W{}$ to its static limit, \ie, $\W{}(1,2) = \W{}(\{\br{1},\sigma_1\},\{\br{2},\sigma_2\}; \omega=0)$.
+Finally, the static approximation is enforced, \ie, $\W{}(1,2) = \W{}(\{\br{1}, \sigma_1, t_1\},\{\br{2}, \sigma_2, t_2\}) \delta(t_1-t_2)$, which corresponds to restricting $\W{}$ to its static limit, \ie, $\W{}(1,2) = \W{}(\{\br{1},\sigma_1\},\{\br{2},\sigma_2\}; \omega=0)$.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{BSE in a finite basis}
@@ -296,10 +300,9 @@ To compute the singlet and triplet BSE excitation energies in a finite basis wit
 		\bY_m	\\
 	\end{pmatrix},
 \end{equation}
-where $\OmBSE{m}$ is the $m$th excitation energy with eigenvector $\T{(\bX_m \, \bY_m)}$, and we have assumed real-valued orbitals.
-
-The matrices $\bA$, $\bB$, $\bX$, and $\bY$ are all of size $OV \times OV$ where $O$ and $V$ are the number of occupied and virtual orbitals, respectively.
-In the following, the index $m$ labels the $OV$ single excitations, $i$, $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, while $p$, $q$, $r$, and $s$ indicate arbitrary orbitals.
+where $\OmBSE{m}$ is the $m$th excitation energy with eigenvector $\T{(\bX_m \, \bY_m)}$, and we have assumed real-valued orbitals $\{\MO{p}(\br{})\}_{1 \le p \le \Norb}$.
+The matrices $\bA$, $\bB$, $\bX$, and $\bY$ are all of size $\Nocc \Nvir \times \Nocc \Nvir$ where $\Nocc$ and $\Nvir$ are the number of occupied and virtual orbitals (\ie, $\Norb = \Nocc + \Nvir$), respectively.
+In the following, the index $m$ labels the $\Nocc \Nvir$ single excitations, $i$, $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, while $p$, $q$, $r$, and $s$ indicate arbitrary orbitals.
 
 In the absence of triplet instabilities, \cite{Dreuw_2005} Eq.~\eqref{eq:LR} is usually transformed into an Hermitian eigenvalue problem of smaller dimension
 \begin{equation}
@@ -324,17 +327,17 @@ where $\eGW{p}$ are the {\GW} quasiparticle energies,
 \label{eq:W}
 	\W{ia,jb}(\omega) = (1 + \delta_{\sigma \sigma^{\prime}}) (ia|jb) 
 	\\
-	+ \sum_m^{OV} [ia|m] [jb|m] \qty(\frac{1}{\omega - \OmRPA{m} + i \eta} + \frac{1}{\omega + \OmRPA{m} - i \eta})
+	+ \sum_m^{\Nocc \Nvir} [ia|m] [jb|m] \qty(\frac{1}{\omega - \OmRPA{m} + i \eta} + \frac{1}{\omega + \OmRPA{m} - i \eta})
 \end{multline}
 are the elements of the screened Coulomb operator, 
 \begin{equation}
-	[pq|m] =  \sum_i^O \sum_a^V (pq|ia) (\bX_m+\bY_m)_{ia}
+	[pq|m] =  \sum_i^{\Nocc} \sum_a^{\Nvir} (pq|ia) (\bX_m+\bY_m)_{ia}
 \end{equation}
 are the screened two-electron integrals, 
 \begin{equation}
 	(pq|rs) = \iint \frac{\MO{p}(\br{}) \MO{q}(\br{}) \MO{r}(\br{}') \MO{s}(\br{}')}{\abs*{\br{} - \br{}'}} \dbr{} \dbr{}',
 \end{equation}
-are the bare two-electron integrals, $\MO{p}(\br{})$ are the molecular orbitals, $\delta_{pq}$ is the Kronecker delta, and
+are the bare two-electron integrals, $\delta_{pq}$ is the Kronecker delta, and
 \begin{equation}
 	\delta_{\sigma \sigma^{\prime}} = 
 	\begin{cases}
@@ -349,7 +352,7 @@ In Eq.~\eqref{eq:W}, $\OmRPA{m}$ are the neutral (direct) RPA excitation energie
 \subsection{Ground- and excited-state BSE energy}
 \label{sec:BSE_energy}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-The key quantity to define here is the total BSE energy as we are going to compute PES following this definition.
+\titou{The key quantity to define here is the total BSE energy as we are going to compute PES following this definition.}
 Although not unique, we propose to define the BSE total energy of the $m$th state as
 \begin{equation}
 \label{eq:EtotBSE}
@@ -379,7 +382,7 @@ As a final remark, we point out that Eq.~\eqref{eq:EtotBSE} can be easily genera
 \section{Computational details}
 \label{sec:comp_details}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-All the preliminary {\GW} calculations performed to obtain the screened Coulomb operator and the quasiparticle energies have been done using a Hartree-Fock (HF) starting point, which is a very sensible choice in the case of the (small) systems that are considered here.
+All the preliminary {\GW} calculations performed to obtain the screened Coulomb operator and the quasiparticle energies have been done using a Hartree-Fock (HF) starting point, which is a very adequate choice in the case of the (small) systems that we consider here.
 Both perturbative {\GW} (or {\GOWO}) \cite{Hybertsen_1985a, Hybertsen_1986} and partially self-consistent {\evGW} \cite{Hybertsen_1986, Shishkin_2007, Blase_2011, Faber_2011} calculations are performed here.
 These will be labeled as BSE@{\GOWO} and BSE@{\evGW}, respectively.
 In the case of {\GOWO}, the quasiparticle energies have been obtained by linearizing the non-linear, frequency-dependent quasiparticle equation.